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Studies in Computational Intelligence

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Gras · Suzuki · GuilletSpagnolo (Eds.)

Statistical Implicative Analysis

The series Studies in Computational Intelligence (SCI) publishes new developments and advances in the various areas of computational intelligence – quickly and with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life science, as well as the methodologies behind them. The series contains monographs, lecture notes and edited volumes in computational intelligence spanning the areas of neural networks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems and hybrid intelligent systems. Critical to both contributors and readers are the short publication time and world-wide distribution – this permits a rapid and broad dissemination of research results.

isbn 978-3-540-78982-6

springer.com

issn 1860-949x

Statistical implicative analysis is a data analysis method created by Régis Gras almost thirty years ago which has a significant impact on a variety of areas ranging from pedagogical and psychological research to data mining. Statistical implicative analysis (SIA) provides a framework for evaluating the strength of implications; such implications are formed through common knowledge acquisition techniques in any learning process, human or artificial. This new concept has developed into a unifying methodology, and has generated a powerful convergence of thought between mathe-maticians, statisticians, psychologists, specialists in pedagogy and last, but not least, computer scientists specialized in data mining.

This volume collects significant research contributions of several rather distinct dis-ciplines that benefit from SIA. Contributions range from psychological and pedagog-ical research, bioinformatics, knowledge management, and data mining.

Régis Gras · Einoshin Suzuki Fabrice Guillet · Filippo Spagnolo (Eds.)

Statistical Implicative AnalysisTheory and Applications

Preface

Statistical implicative analysis is a data analysis method created by RégisGras almost thirty years ago which has a signi�cant impact on a variety of ar-eas ranging from pedagogical and psychological research to data mining. Thisnew concept has developed into a unifying methodology, and has generated apowerful convergence of thought between mathematicians, statisticians, psy-chologists, specialists in pedagogy and last, but not least, computer scientistsspecialized in data mining.

Statistical implicative analysis (SIA) provides a framework for evaluatingthe strength of implications; such implications are formed through commonknowledge acquisition techniques in any learning process, human or arti�cial.Therefore, the epistemological interest of SIA is, in my opinion, of universalinterest for researchers. In many applications implications appear as �rules�and, as it is often the case, rules have exceptions. SIA provides a powerfulinstrument for quantifying the quality of a rule taking into account the real-ity of these exceptions. Many applications, especially in data mining, extractlarge sets of rules that are impossible to assimilate by humans and used e�-ciently in decision processes. Therefore, it is important to develop measuresof interestingness for these rules and the success of SIA-based techniques inthis direction is indisputable.

This volume collects signi�cant research contributions of several ratherdistinct disciplines that bene�t from SIA. Contributions range from psycho-logical and pedagogical research, bioinformatics, knowledge management, anddata mining.

The �rst applications of SIA were in the realm of didactics and this �eldis richly represented here by several contributions that focus on such diverseproblems as didactics of algebra and geometry, the teaching of functions rep-resentations and graphing, Bayesian inference, and student representations ofphysical activities.

Interesting data mining applications authored by leading researchers inthe �eld range from applying SIA in the study of rules produced by decisiontrees, association rules generated by the analysis of transactional data, tempo-

VI Preface

ral rules, measures of interestingness for various types of rules, and hierarchicalorganization of rules. A novel method for analyzing DNA microarrays is for-mulated using SIA concepts. Furthermore, applications of SIA to the study ofontologies and textual taxonomies, as well as applications to fuzzy knowledgediscovery are also included.

We have here a new volume that con�rms the validity of a novel andpowerful statistical methodology, though many convincing applications. Thecontributors have done a masterful job of exposition.

After reading this book, I have in mind a few applications of SIA in my ownresearch. I am convinced that the readers will �nd this volume as stimulatingas I did.

Boston, Prof. Dan A. SimoviciSeptember, 2007 Department of Computer Science

University of Massachusetts at Boston University

Preface VII

Review Committee

All published chapters have been reviewed by at least 2 referees.

� Saddo Ag Almouloud (University of Sao Paulo, Brazil)� Carmen Batanero (University of Grenada)� Hans Bock (Aachen University, Germany)� Henri Briand (LINA, University of Nantes, France)� Guy Brousseau (University of Bordeaux 3, France)� Alex Freitas (University of Kent, UK)� Athanasios Gagatsis (University of Chyprius)� Robin Gras (University of Windsor, Canada)� Howard Hamilton (University of Regina, Canada)� Jiawei Han (University of Illinois, USA)� David J. Hand (Imperial College, London, UK)� André Hardy (University of Namur, Belgium)� Robert Hilderman (University of Regina, Canada)� Yves Kodrato� (LRI, University of Paris-Sud, France)� Pascale Kuntz (LINA, University of Nantes, France)� Ludovic Lebart (ENST, Paris, France)� Amédéo Napoli (LORIA, University of Nancy, France)� Maria-Gabriella Ottaviani (University of Roma, Italy)� Balaji Padmanabhan (University of Pennsylvania, USA)� Jean-Paul Rasson (University of Namur, Belgium)� Jean-Claude Régnier (University of Lyon 2, France)� Gilbert Ritschard (Geneve University, Switzerland)� Lorenza Saitta (University of Piemont, Italy)� Gilbert Saporta (CNAM, Paris, France)� Dan Simovici (University of Massachusetts Boston, USA)� Djamel Zighed (ERIC, University of Lyon 2, France)

Associated Reviewers

Nadja Maria Acioly-Régnier,Angela Alibrandi,Jérôme Azé,Maurice Bernadet,Julien Blanchard,Catherine-Marie Chiocca,Raphaël Couturier,Stéphane Daviet,Jérôme David,

Carmen Diaz,Pablo Gregori,Alain Kuzniak,Eduardo Lacasta,Dominique Lahanier-Reuter,Stéphane Lallich,Letitzia La Tona,Patrick Leconte,Rémi Lehn,

Philippe Lenca,Elsa Malisani,Rajesh Natajaran,Pilar Orús,Gérard Ramstein,Ansaf Salleb,Aldo Scimone,Benoît Vaillant,Ingrid Verscheure

Manuscript coordinator

Bruno Pinaud (LINA, University of Nantes, France)

VIII Preface

Acknowledgments

The editors would like to thank the chapter authors for their insights andcontributions to this book.

The editors would also like to acknowledge the members of the review com-mittee and the associated referees for their involvement in the review processof the book, and without whose support the book would not have been satis-factorily completed.

A special thank goes to H. Briand for his encouragements.

Thanks also to J. Blanchard who has managed the cyberchair web site.

Finally, we thank Springer and the publishing team, and especially T.Ditzinger and J. Kacprzyk, for their con�dence in our project.

Nantes, Régis GrasDecember 2007 Einoshin Suzuki

Fabrice GuilletFilippo Spagnolo

Contents

IntroductionRégis Gras, Einoshin Suzuki, Fabrice Guillet, Filippo Spagnolo . . . . . . . . . 1

Part I Methodology and concepts for SIA

An overview of the Statistical Implicative Analysis (SIA)developmentRégis Gras, Pascale Kuntz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHIC:Cohesive Hierarchical Implicative Classi�cationRaphaël Couturier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Assessing the interestingness of temporal rules withSequential Implication IntensityJulien Blanchard, Fabrice Guillet, Régis Gras . . . . . . . . . . . . . . . . . . . . . . . . 55

Part II Application to concept learning in education, teaching, anddidactics

Student's Algebraic Knowledge Modelling: Algebraic Contextas Cause of Student's ActionsMarie-Caroline Croset, Jana Trgalova, Jean-François Nicaud . . . . . . . . . . 75

The graphic illusion of high school studentsEduardo Lacasta, Miguel R. Wilhelmi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Implicative networks of student's representations of PhysicalActivitiesCatherine-Marie Chiocca, Ingrid Verscheure . . . . . . . . . . . . . . . . . . . . . . . . . 119

X Contents

A comparison between the hierarchical clustering of variables,implicative statistical analysis and con�rmatory factoranalysisIliada Elia, Athanasios Gagatsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Implications between learning outcomes in elementarybayesian inferenceCarmen Díaz, Inmaculada de la Fuente, Carmen Batanero . . . . . . . . . . . . 163

Personal Geometrical Working Space: a Didactic andStatistical ApproachAlain Kuzniak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Part III A methodological answer in various applicationframeworks

Statistical Implicative Analysis of DNA microarraysGerard Ramstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

On the use of Implication Intensity for matching ontologiesand textual taxonomiesJérôme David, Fabrice Guillet, Henri Briand, Régis Gras . . . . . . . . . . . . . 227

Modelling by Statistic in Research of Mathematics EducationElsa Malisani and Aldo Scimone and Filippo Spagnolo . . . . . . . . . . . . . . . . 247

Didactics of Mathematics and Implicative Statistical AnalysisDominique Lahanier-Reuter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Using the Statistical Implicative Analysis for ElaboratingBehavioral ReferentialsStéphane Daviet, Fabrice Guillet, Henri Briand, Serge Baquedano,Vincent Philippé, Régis Gras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Fictitious Pupils and Implicative Analysis: a Case StudyPilar Orús, Pablo Gregori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Identifying didactic and sociocultural obstacles toconceptualization through Statistical Implicative AnalysisNadja Maria Acioly-Régnier, Jean-Claude Régnier . . . . . . . . . . . . . . . . . . . . 347

Part IV Extensions to rule interestingness in data mining

Pitfalls for Categorizations of Objective InterestingnessMeasures for Rule DiscoveryEinoshin Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Contents XI

Inducing and Evaluating Classi�cation Trees with StatisticalImplicative CriteriaGilbert Ritschard, Vincent Pisetta, Djamel A. Zighed . . . . . . . . . . . . . . . . . 397

On the behavior of the generalizations of the intensity ofimplication: A data-driven comparative studyBenoît Vaillant, Stéphane Lallich, Philippe Lenca . . . . . . . . . . . . . . . . . . . . 421

The TVpercent principle for the counterexamples statisticRicco Rakotomalala, Alain Morineau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

User-System Interaction for Redundancy-Free KnowledgeDiscovery in DataRémi Lehn, Henri Briand, Fabrice Guillet . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Fuzzy Knowledge Discovery Based on Statistical ImplicationIndexesMaurice Bernadet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

About the editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509